The local circular law II: the edge case

Paul Bourgade Harvard University Horng-Tzer Yau Harvard University Jun Yin University of Wisconsin-Madison

Probability mathscidoc:1608.28010

Probability Theory & Related Fields, 159, (3), 619-660, 2012
In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ \e}$ for non-Hermitian random matrices at any point $z \in \C$ with ||z|−1|>c for any c>0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $|z|-1=\oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $|z|-1=\oo(1)$ up to scale $N^{-1/4+ \e}$.
local circular law, universality
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@inproceedings{paul2012the,
  title={The local circular law II: the edge case},
  author={Paul Bourgade, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823135700538512404},
  booktitle={Probability Theory & Related Fields},
  volume={159},
  number={3},
  pages={619-660},
  year={2012},
}
Paul Bourgade, Horng-Tzer Yau, and Jun Yin. The local circular law II: the edge case. 2012. Vol. 159. In Probability Theory & Related Fields. pp.619-660. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823135700538512404.
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